Deck 16: Mathematics Problems: Differential Equations and Linear Algebra

Question 1

The solution of $y ^ { \prime \prime } - 4 y ^ { \prime } + 13 y = 0$ is

A) $y = c _ { 1 } e ^ { - 2 x } \cos ( 3 x ) + c _ { 2 } e ^ { - 2 x } \sin ( 3 x )$
B) $y = c _ { 1 } e ^ { - 2 x } \cos ( 3 x ) + c _ { 2 } e ^ { 2 x } \sin ( 3 x )$
C) $y = c _ { 1 } e ^ { 2 x } \cos ( 3 x ) + c _ { 2 } e ^ { 2 x } \sin ( 3 x )$
D) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 3 x }$
E) $y = c _ { 1 } \cos ( 3 x ) + c _ { 2 } \sin ( 3 x )$

Answer 1

$y = c _ { 1 } e ^ { 2 x } \cos ( 3 x ) + c _ { 2 } e ^ { 2 x } \sin ( 3 x )$

Question 2

The solution of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ is

A) $y = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 3 x }$
B) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } x e ^ { 3 x }$
C) $y = c _ { 1 } e ^ { - 2 x } + c _ { 2 } x e ^ { - 3 x }$
D) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 3 x }$
E) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { - 3 x }$

Answer 2

$y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 3 x }$

Question 3

The solution of $x y ^ { \prime } = ( x + 1 ) y ^ { 2 }$

A) $y = 1 / ( x + \ln x + c )$
B) $y = 1 / ( x - \ln x + c )$
C) $y = - c / ( x + \ln x )$
D) $y = - c / ( x - \ln x )$
E) $y = - 1 / ( x + \ln x + c )$

Answer 3

$y = - 1 / ( x + \ln x + c )$

Question 4

The correct form of the particular solution of $y ^ { \prime \prime } - 2 y ^ { \prime } = x + e ^ { x }$ is

A) $y _ { p } = A x + B + C e ^ { x }$
B) $y _ { p } = \left( A x + B + C e ^ { x } \right) x$
C) $y _ { p } = A x ^ { 2 } + B + C e ^ { x }$
D) $y _ { p } = A x ^ { 2 } + B x + C e ^ { x }$
E) $y _ { p } = A x + B + C x e ^ { x }$

Question 5

The solution of $y ^ { \prime \prime } + y = \tan x$ is

A) $y = c _ { 1 } \cos x + c _ { 2 } \sin x + \cos x \ln | \sec x + \tan x |$
B) $y = c _ { 1 } \cos x + c _ { 2 } \sin x - \cos x \ln | \sec x + \tan x |$
C) $y = c _ { 1 } \cos x + c _ { 2 } \sin x + \cos x \ln | \sec x |$
D) $y = c _ { 1 } \cos x + c _ { 2 } \sin x - \cos x \ln | \tan x |$
E) $y = c _ { 1 } \cos x + c _ { 2 } \sin x - \cos x \ln | \sec x - \tan x |$

Question 6

The solution of $x ^ { 2 } y d x + \left( x ^ { 3 } / 3 + y \right) d y = 0$ is

A) $x ^ { 3 } y / 3 + y ^ { 2 } / 2$
B) $x ^ { 3 } y / 3 - y ^ { 2 } / 2$
C) $x ^ { 3 } y / 3 + y ^ { 2 } / 2 - c$
D) $x ^ { 3 } y / 3 + y ^ { 2 } / 2 = c$
E) $x ^ { 3 } y / 3 - y ^ { 2 } / 2 = c$

Question 7

The solution of $y ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = \cos x$ is

A) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( 5 \sin x + 3 \cos x ) / 34$
B) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( - 5 \sin x + 3 \cos x ) / 34$
C) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( - 5 \cos x - 3 \sin x ) / 34$
D) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( 5 \cos x + 3 \sin x ) / 34$
E) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 4 x } + ( - 5 \cos x + 3 \sin x ) / 34$

Question 8

The solution of $y ^ { \prime } - y = x$ is

A) $y = x - 1 + c e ^ { x }$
B) $y = - x + 1 + c e ^ { x }$
C) $y = - x - 1 + c e ^ { x }$
D) $y = - x - 1 + c e ^ { - x }$
E) $y = x + 1 + c e ^ { - x }$

Question 9

The auxiliary equation of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ is

A) $m ^ { 2 } - 5 m - 6 = 0$
B) $m ^ { 2 } - 5 m + 6 = 0$
C) $m ^ { 2 } - 5 m + 6 = 1$
D) $m ^ { 2 } - 5 m + 6$
E) $m ^ { 2 } - 5 m - 6$

Question 10

The solution of $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } - 3 y = 0$ is

A) $y = c _ { 1 } x + c _ { 2 } x ^ { - 3 }$
B) $y = c _ { 1 } x ^ { - 1 } + c _ { 2 } x ^ { 3 }$
C) $y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - 3 x }$
D) $y = c _ { 1 } e ^ { - x } + c _ { 2 } e ^ { 3 x }$
E) $y = c _ { 1 } x + c _ { 2 } x ^ { 3 }$

Question 11

The solution of $y ^ { \prime \prime } + 4 y ^ { \prime } + 4 y = 0$ is

A) $y = c _ { 1 } e ^ { - 2 x } + c _ { 2 } x e ^ { - 2 x }$
B) $y = c _ { 1 } e ^ { - 2 x } + c _ { 2 } e ^ { - 2 x }$
C) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 2 x }$
D) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } x e ^ { 2 x }$
E) $y = c _ { 1 } e ^ { 2 x } + c _ { 2 } e ^ { 4 x }$

Question 12

The solution of $y ^ { \prime \prime } - 2 y ^ { \prime } = x + e ^ { x }$ is

A) $y = c _ { 1 } + c _ { 2 } e ^ { 2 x } - x ^ { 2 } / 4 - x / 4 - e ^ { x }$
B) $y = c _ { 1 } + c _ { 2 } e ^ { 2 x } - x ^ { 2 } / 4 - x / 4 + e ^ { x }$
C) $y = c _ { 1 } + c _ { 2 } e ^ { 2 x } + x ^ { 2 } / 4 - x / 4 - e ^ { x }$
D) $y = c _ { 1 } + c _ { 2 } e ^ { 2 x } + x ^ { 2 } / 4 + x / 4 - e ^ { x }$
E) $y = c _ { 1 } + c _ { 2 } e ^ { 2 x } + x ^ { 2 } / 4 + x / 4 + e ^ { x }$

Question 13

The correct form of the particular solution of $y ^ { \prime \prime } - 2 y ^ { \prime } + y = e ^ { x }$ is

A) $y _ { p} = A e ^ { x }$
B) $y _ { p } = A x e ^ { x }$
C) $y _ { p } = A x ^ { 2 } e ^ { x }$
D) $y _ { p } = A x ^ { 3 } e ^ { x }$
E) none of the above

Question 14

A frozen chicken at $0 ^ { \circ } \mathrm { C }$ is taken out of the freezer and placed on a table at $20 ^ { \circ } \mathrm { C }$ . One hour later the temperature of the chicken is $18 ^ { \circ } \mathrm { C }$ . The mathematical model for the temperature $T ( t )$ as a function of time $t$ is (assuming Newton's law of warming)

A) $\frac { d T } { d t } = k T , T ( 0 ) = 0 , T ( 1 ) = 18$
B) $\frac { d T } { d t } = k ( T - 20 ) , T ( 0 ) = 0 , T ( 1 ) = 18$
C) $\frac { d T } { d t } = ( T - 20 ) , T ( 0 ) = 0 , T ( 1 ) = 18$
D) $\frac { d T } { d t } = T , T ( 0 ) = 0 , T ( 1 ) = 18$
E) $\frac { d T } { d t } = k ( T - 18 ) , T ( 0 ) = 0 , T ( 1 ) = 18$

Question 15

The solution of $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } = 0$ is

A) $y = c _ { 1 } + c _ { 2 } x ^ { - 1 }$
B) $y = c _ { 1 } \ln x + c _ { 2 } x ^ { - 1 }$
C) $y = c _ { 1 } + c _ { 2 } \ln x$
D) $y = c _ { 1 } + c _ { 2 } x$
E) $y = c _ { 1 } + c _ { 2 } x ^ { - 2 }$

Question 16

A 2-pound weight is hung on a spring and stretches it 1/2 foot. The mass spring system is then put into motion in a medium offering a damping force numerically equal to the velocity. If the mass is pulled down 4 inches from equilibrium and released, the initial value problem describing the position, $x ( t )$ , of the mass at time $t$ is

A) $x ^ { \prime \prime } - 16 x ^ { t } + 64 x = 0 , x ( 0 ) = 4 , x ^ { \prime } ( 0 ) = 0$
B) $x ^ { \prime \prime } + 16 x ^ { t } + 64 x = 0 , x ( 0 ) = 4 , x ^ { \prime } ( 0 ) = 0$
C) $x ^ { \prime \prime } - 16 x ^ { \prime } + 64 x = 0 , x ( 0 ) = 1 / 3 , x ^ { \prime } ( 0 ) = 0$
D) $x ^ { \prime \prime } + 16 x ^ { \prime } + 64 x = 0 , x ( 0 ) = 1 / 3 , x ^ { \prime } ( 0 ) = 0$
E) $x ^ { \prime \prime } + 64 x = 16 , x ( 0 ) = 1 / 3 , x ^ { \prime } ( 0 ) = 0$

Question 17

In the previous problem, the solution of the differential equation is

A) $T = C e ^ { k t }$
B) $T = C e ^ { - k t }$
C) $T = 20 + C e ^ { k t }$
D) $T = 20 + C e ^ { - k t }$
E) $T = 18 + C e ^ { k t }$

Question 18

In the previous two problems, the solution for the temperature is

A) $T ( t ) = 20 - 20 e ^ { - 230 t }$
B) $T ( t ) = 20 - 20 e ^ { 230 t }$
C) $T ( t ) = 18 - 18 e ^ { - 230 t }$
D) $T ( t ) = 18 - 18 e ^ { 230 t }$
E) $T ( t ) = 18 e ^ { - 230 t }$

Question 19

In the previous problem, the solution for the position, $x ( t )$ , is

A) $x = \left( e ^ { 8 t } + 8 t e ^ { 8 t } \right) / 3$
B) $x = \left( e ^ { - 8 t } + 8 t e ^ { - 8 t } \right) / 3$
C) $x = \cos ( 8 t ) / 12 + 1 / 4$
D) $x = \left( 4 e ^ { - 8 t } + 32 t e ^ { - 8 t } \right)$
E) $x = \left( 4 e ^ { 8 t } - 32 t e ^ { 8 t } \right)$

Question 20

The solution of $y y ^ { \prime \prime } = \left( y ^ { \prime } \right) ^ { 2 }$ is

A) $y = c _ { 1 } e ^ { C _ { 2 } x }$
B) $y \ln \left( c _ { 1 } \right) + y \ln y - y = - x + c _ { 2 }$
C) $y = c _ { 1 } x + c _ { 2 }$
D) $y \ln y - y = c _ { 1 } x + c _ { 2 }$
E) $y = c _ { 1 } \ln \left( c _ { 2 } x \right)$

Question 21

Using Laplace transform methods, the solution of $y ^ { \prime } + y = 2 \sin t , y ( 0 ) = 1$ is (Hint: the previous problem might be useful.)

A) $y = 2 e ^ { - t } + \sin t + \cos t$
B) $y = e ^ { t } + e ^ { - t } - \sin t - \cos t$
C) $y = 2 e ^ { - t } - \sin t - \cos t$
D) $y = 2 e ^ { - t } + \sin t - \cos t$
E) $y = e ^ { t } + e ^ { - t } + \sin t - \cos t$

Question 22

Let $A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)$ . Then $e ^ { A t } =$

A) $I + A t$
B) $I + A t + A ^ { 2 } t ^ { 2 } / 2$
C) $A t + A ^ { 2 } t ^ { 2 } / 2$
D) $\left( \begin{array} { c c } e ^ { 2 t } & 0 \\0 & e ^ { - 3 t }\end{array} \right)$
E) $\left( \begin{array} { c c } e ^ { 2 t - 1 } - 1 & 0 \\0 & e ^ { - 3 t } - 1\end{array} \right)$

Question 23

The solution of the previous problem is

A) $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 24 E I )$
B) $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 12 E I )$
C) $y = w _ { 0 } E I \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) + 15 t ^ { 2 } - 2 t ^ { 3 } \right] / 24$
D) $y = w _ { 0 } E I \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) - 15 t ^ { 2 } - 2 t ^ { 3 } \right] / 12$
E) $y = w _ { 0 } \left[ 4 ( t - 5 ) ^ { 3 } u ( t - 5 ) - 15 t ^ { 2 } - 2 t ^ { 3 } \right] / ( 24 E I )$

Question 24

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + 2 y ^ { \prime } + y = e ^ { - t } , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

A) $y = e ^ { t } + t e ^ { t } + t ^ { 2 } e ^ { t } / 2$
B) $y = e ^ { - t } + t e ^ { - t } + t ^ { 2 } e ^ { - t } / 2$
C) $y = e ^ { t } - t e ^ { t } + t ^ { 2 } e ^ { t } / 2$
D) $y = e ^ { - t } - t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$
E) $y = e ^ { - t } + t e ^ { - t } - t ^ { 2 } e ^ { - t } / 2$

Question 25

In the previous problem, the exact solution of the initial value problem is

A) $y = \left( e ^ { 2 x } - 1 \right) / \left( e ^ { 2 x } + 1 \right)$
B) $y = \left( e ^ { 2 x } + 1 \right) / \left( e ^ { 2 x } - 1 \right)$
C) $y = \left( e ^ { - 2 x } - 1 \right) / \left( e ^ { - 2 x } + 1 \right)$
D) $y = - \left( e ^ { - 2 x } + 1 \right) / \left( e ^ { - 2 x } - 1 \right)$
E) $y = - \left( e ^ { 2 x } - 1 \right) / \left( e ^ { 2 x } + 1 \right)$

Question 26

The eigenvalues of the matrix $A = \left( \begin{array} { c c } 1 & - 2 \\1 & 4\end{array} \right)$ are

A) $( 5 \pm \sqrt { 17 } ) / 2$
B) $( - 5 \pm \sqrt { 17 } ) / 2$
C) 1, 2
D) 2, 3
E) 2, 2

Question 27

The solution of $$X ^ { \prime } = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) \mathrm { X }$$ is

A) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
B) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
C) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
D) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { l } 0 \\1 \\1\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { l } 0 \\1 \\1\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$
E) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 1 \\0 \\0\end{array} \right) e ^ { 3 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1 \\0\end{array} \right) e ^ { 2 t } + c _ { 3 } \left[ \left( \begin{array} { c } 1 \\- 1 \\0\end{array} \right) t e ^ { 2 t } + \left( \begin{array} { l } 0 \\1 \\0\end{array} \right) e ^ { 2 t } \right]$$

Question 28

The solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { c c } 1 & - 2 \\1 & 4\end{array} \right) \mathbf { X }$ is

A) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 2 \\1\end{array} \right) e ^ { - 2 t } + c _ { 2 } \left( \begin{array} { l } 1 \\1\end{array} \right) e ^ { - 3 t }$$
B) $$X = c _ { 1 } \left( \begin{array} { l } 2 \\1\end{array} \right) e ^ { - 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { - 3 t }$$
C) $$X = c _ { 1 } \left( \begin{array} { c } - 2 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { 3 t }$$
D) $$X = c _ { 1 } \left( \begin{array} { c } 1 \\- 2\end{array} \right) e ^ { 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { 3 t }$$
E) $$X = c _ { 1 } \left( \begin{array} { c } 2 \\- 1\end{array} \right) e ^ { 2 t } + c _ { 2 } \left( \begin{array} { c } 1 \\- 1\end{array} \right) e ^ { 3 t }$$

Question 29

A particular solution of $$\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right) \mathbf { X } + \left( \begin{array} { l } 2 \\t\end{array} \right)$$ is

A) $$\mathrm { X } _ { \boldsymbol { p } } = \left( \begin{array} { c } 2 t + 2 \\- t + 3\end{array} \right)$$
B) $$\mathrm { X } _ { p } = \left( \begin{array} { c } - 2 t + 2 \\- t + 3\end{array} \right)$$
C) $$X _ { p } = \left( \begin{array} { c } - 2 t + 2 \\- t - 3\end{array} \right)$$
D) $$\mathrm { X } _ { y } = \left( \begin{array} { c } - 2 t + 2 \\t + 3\end{array} \right)$$
E) $$X _ { y } = \left( \begin{array} { c } - 2 t - 2 \\- t - 3\end{array} \right)$$

Question 30

Using the improved Euler method with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

A) $y _ { 1 } = 0.095$
B) $y _ { 1 } = 0.995$
C) $y _ { 1 } = 0.95$
D) $y _ { 1 } = 0.00995$
E) $y _ { 1 } = 0.0995$

Question 31

Using the convolution theorem, we find that $\mathcal { L } ^ { - 1 } \left\{ 1 / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right) \right\} =$

A) $\left( e ^ { - t } + \sin t - \cos t \right) / 2$
B) $\left( e ^ { t } + \sin t - \cos t \right) / 2$
C) $\left( e ^ { - t } + \sin t + \cos t \right) / 2$
D) $\left( e ^ { t } - \sin t - \cos t \right) / 2$
E) $\left( e ^ { - t } - \sin t - \cos t \right) / 2$

Question 32

The solution of $$\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right) \mathbf { X }$$ is

A) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l } 2 \\1\end{array} \right) \cos t - \left( \begin{array} { c } 0 \\- 1\end{array} \right) \sin t \right] + c _ { 2 } \left[ \left( \begin{array} { l } 2 \\1\end{array} \right) \sin t + \left( \begin{array} { c } 0 \\- 1\end{array} \right) \cos t \right]$$
B) $$X = c _ { 1 } \left( \begin{array} { l } 1 \\0\end{array} \right) e ^ { \sqrt { 3 } t } + c _ { 2 } \left( \begin{array} { l } 0 \\1\end{array} \right) e ^ { - \sqrt { 3 } t }$$
C) $$\mathbf { X } = c _ { 1 } \left( \begin{array} { l } 1 \\0\end{array} \right) e ^ { t } + c _ { 2 } \left( \begin{array} { l } 0 \\1\end{array} \right) e ^ { - t }$$
D) $$\begin{array} { l } \mathrm { X } = c _ { 1 } \left[ \left( \begin{array} { l } 2 \\1\end{array} \right) \cos ( \sqrt { 3 } t ) - \left( \begin{array} { c } 0 \\- 1\end{array} \right) \sin ( \sqrt { 3 } t ) \right] + \\c _ { 2 } \left[ \left( \begin{array} { l } 2 \\1\end{array} \right) \sin ( \sqrt { 3 } t ) + \left( \begin{array} { c } 0 \\- 1\end{array} \right) \cos ( \sqrt { 3 } t ) \right]\end{array}$$
E) $$\mathbf { X } = c _ { 1 } \left[ \left( \begin{array} { l } 2 \\1\end{array} \right) \cos ( 2 t ) - \left( \begin{array} { c } 0 \\- 1\end{array} \right) \sin ( 2 t ) \right] + c _ { 2 } \left[ \left( \begin{array} { l } 2 \\1\end{array} \right) \sin ( 2 t ) + \left( \begin{array} { c } 0 \\- 1\end{array} \right) \cos ( 2 t ) \right]$$

Question 33

In the previous two problems, the error in the improved Euler method at $x = 0.1$ is

A) 0.00467
B) 0.000168
C) 0.870
D) 0.895
E) 0.0897

Question 34

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + y = \delta ( t - \pi ) , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0$ is

A) $y = \sin t + \sin ( t - \pi ) \boldsymbol { u } ( t - \pi )$
B) $y = \sin t - \cos ( t - \pi ) u ( t - \pi )$
C) $y = \cos t + \sin ( t - \pi ) u ( t - \pi )$
D) $y = \cos t + \cos ( t - \pi ) u ( t - \pi )$
E) $y = \cos t - \sin ( t - \pi ) u ( t - \pi )$

Question 35

A uniform beam of length 10 has a concentrated load $w _ { 0 }$ at $x = 5$ . It is embedded at both ends. The boundary value problem for the deflections, $y ( x )$ , for this system is

A) $y ^ { \prime \prime \prime \prime } = E L w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$
B) $y ^ { \prime \prime } = E l w _ { 0 } \delta ( x - 10 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$
C) $E l y ^ { \prime \prime } = w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$
D) $E L y ^ { \prime \prime \prime } = w _ { 0 } \delta ( x - 5 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$
E) $E L y ^ { \prime \prime \prime \prime } = w _ { 0 } \delta ( x - 10 ) , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 10 ) = 0 , y ^ { \prime } ( 10 ) = 0$

Question 36

The eigenvalues of the matrix $$A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right)$$ are

A) 1, 2, 3
B) 2, 2, 3
C) 1, 2, 2
D) $- 2 , - 2,3$
E) $- 1 , - 2,3$

Question 37

Using power series methods, the solution of $2 x y ^ { \prime \prime } + y ^ { \prime } + 2 y = 0$ is

A) $$\begin{array} { l } y = c _ { 0 } \sum _ { n - 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) + \\c _ { 1 } x ^ { 1 / 2 } \sum _ { n - 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) )\end{array}$$
B) $$\begin{array} { l } y = c _ { 0 } \sum _ { n - 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) + \\c _ { 1 } x ^ { 1 / 2 } \left[ 1 + \sum _ { n - 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$$
C) $$\begin{array} { l } y = c _ { 0 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\c _ { 1 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$$
D) $$\begin{array} { l } y = c _ { 0 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\c _ { 1 } x ^ { 1 / 2 } \left[ 1 + \sum _ { n = 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$$
E) $$\begin{array} { l } y = \left[ 1 + \sum _ { n - 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 1 \cdot 3 \cdots ( 2 n - 1 ) ) ) \right] + \\x ^ { 1 / 2 } \left[ 1 + \sum _ { n - 1 } ^ { \infty } ( - 2 ) ^ { n } x ^ { n } / ( n ! ( 3 \cdot 5 \cdots ( 2 n + 1 ) ) ) \right]\end{array}$$

Question 38

Using power series methods, the solution of $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ is

A) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / n ! \right]$
B) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n - 1 } ^ { \infty } x ^ { n } / ( n ! ( n + 1 ) ) \right]$
C) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n - 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
D) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n - 1 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
E) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n - 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$

Question 39

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$ is

A) $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$
B) $\lambda = n \pi / 2 , y = \cos ( n \pi x / 2 ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } \pi ^ { 2 } / 4 , y = \sin ( n \pi x / 2 ) , n = 1,2,3 , \ldots$
D) $\lambda = ( 2 n - 1 ) \pi / 2 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$
E) $\lambda = ( 2 n - 1 ) ^ { 2 } \pi ^ { 2 } / 4 , y = \cos ( ( 2 n - 1 ) \pi x / 2 ) , n = 1,2,3 , \ldots$

Question 40

The eigenvalues of the matrix $$A = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right)$$ are

A) $\pm \sqrt { 3 }$
B) $\pm \sqrt { 3 } i$
C) $\pm 1$
D) $\pm 2 i$
E) $\pm$

Question 41

In the previous problem, the error in the classical Runge-Kutta method at $x = 0.1$ is (Hint: see the previous five problems.)

A) 0.0008
B) 0.00008
C) 0.00000008
D) 0.000008
E) 0.0000008

Question 42

The eigenvalues of the matrix $$A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right)$$ are

A) $$\left( \begin{array} { l } 1 \\0 \\0\end{array} \right)$$
B) $$\left( \begin{array} { c } 0 \\1 \\- 1\end{array} \right)$$
C) $$\left( \begin{array} { c } 1 \\- 1 \\0\end{array} \right)$$
D) $$\left( \begin{array} { l } 0 \\1 \\1\end{array} \right)$$
E) $$\left( \begin{array} { l } 0 \\1 \\0\end{array} \right)$$

Question 43

The differential equation $y ^ { \prime } = x ^ { 2 } y ^ { 2 }$ is Select all that apply.

A) linear
B) separable
C) exact
D) non-linear
E) Bernoulli

Question 44

Consider the boundary-value problem $y ^ { \prime \prime } - 4 y ^ { \prime } + 3 y = x , y ( 0 ) = 1 , y ( 1 ) = 2$ . Replace the derivatives with central differences with a step size of $h = 1 / 4$ . The resulting equations are

A) $12 y _ { i + 1 } - 29 y _ { i } - 24 y _ { i - 1 } = x _ { i }$
B) $12 y _ { i + 1 } + 17 y _ { i } + 12 y _ { i - 1 } = x _ { i }$
C) $8 y _ { i + 1 } + 17 y _ { i } + 12 y _ { i - 1 } = x _ { i }$
D) $8 y _ { i + 1 } - 29 y _ { i } + 24 y _ { i - 1 } = x _ { i }$
E) $8 y _ { i + 1 } + 29 y _ { i } - 24 y _ { i - 1 } = x _ { i }$

Question 45

Using the classical Runge-Kutta method of order 4 with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 - y ^ { 2 } , y ( 0 ) = 0$ at $x = 0.1$ is

A) 0.099588
B) 0.099668
C) 0.099688
D) 0.099768
E) 0.099788

Question 46

The differential equation $x d y - y d x = 0$ is Select all that apply.

A) linear
B) separable
C) exact
D) non-linear
E) Bernoulli

Question 47

The differential equation $y ^ { \prime } + y = x ^ { 2 }$ is Select all that apply.

A) linear
B) separable
C) exact
D) non-linear
E) Bernoulli

Question 48

The solution of the system in the previous problem is

A) $y _ { 1 } = 1.228 , y _ { 2 } = 1.482 , y _ { 3 } = 1.753$
B) $y _ { 1 } = 1.228 , y _ { 2 } = 1.646 , y _ { 3 } = 1.753$
C) $y _ { 1 } = 1.126 , y _ { 2 } = 1.646 , y _ { 3 } = 2.903$
D) $y _ { 1 } = 1.126 , y _ { 2 } = 1.786 , y _ { 3 } = 2.903$
E) $y _ { 1 } = 1.016 , y _ { 2 } = 1.786 , y _ { 3 } = 2.903$